Method for facilitating monitoring, in the course of time, of the evolution of physical states in an underground formation

ABSTRACT

Method for facilitating monitoring, in the course of time, of the evolution of the physical states of a zone of an underground formation such as a reservoir by using and interpreting 4D seismic data.  
     The method uses seismic pattern recognition, more particularly a fuzzy discriminant analysis technique allowing to integrate the uncertainties on the seismic measurements in the interpretation. The uncertainties taken into account are evaluated on parts of seismic traces obtained during successive seismic surveys, corresponding to zones of the underground formation (outside the reservoir) where the physical states monitored have undergone no significant change.  
     Application: monitoring of the evolution of a hydrocarbon reservoir during production for example.

FIELD OF THE INVENTION

[0001] The present invention relates to a method for facilitating monitoring, in the course of time, of the evolution of the physical states of an underground formation by using and interpreting 4D seismic data.

BACKGROUND OF THE INVENTION

[0002] Various aspects of the prior art in the sphere considered are described for example in the following publications:

[0003] Dumay, J., Fournier, F., 1988, “Multivariate statistical analyses applied to seismic facies recognition”, Geophysics, 53, n^(o)9, pp. 1151-1159;

[0004] Sonneland, L. et al, 1997, <<Seismic reservoir monitoring on Gullfalks>>, The Leading Edge, 16, n^(o)9, pp. 1247-1252,

[0005] Ross C. et al <<Inside the Crossequalization Blackbox >>, The Leading Edge, 15:11, 1996, pp. 123 3-1240;

[0006] Kolmogorov A. N., 1950, Foundation of the Theory of Probability; Chelsea Publ. Co., New York;

[0007] Moore R. E., 1969, Interval Analysis: Prenctice-Hall, Englewood Cliffs;

[0008] Walley P., 1991, Statistical Reasoning with Imprecise Probabilities: Monographs on Statistics and Applied Probabilities n. 42, Chapman and Hall, London;

[0009] as well as in patents FR-2,768,818 and FR-EN-00/11,618.

[0010] Seismic measurements are conventionally used to provide additional information, in relation to drilling data, on the variations of the subsoil formations: lithologic, petrophysical or fluid saturation variations. In particular, within the scope of hydrocarbon reservoir production, it has become quite frequent to record seismic measurements repeatedly and then to interpret the seismic measurement variations in connection with the saturation and pressure variations due to reservoir production phenomena. This interpretation is often carried out by means of statistical pattern recognition techniques allowing to classify the seismic events into various categories representing the different physical states of the reservoir. These approaches are for example described in the publication by Dumay, J., Fournier, F. (1988). Their application to the interpretation of repeated seismic surveys is for example described in the publication by Sonneland, L., et al. (1997).

[0011] One difficulty concerning interpretation of repeated (or 4D) seismic surveys is that the measurement is not perfectly repetitive. Thus, even in zones of the subsoil that are not affected by the production of the reservoir, and whose seismic response should remain unchanged in the course of time, seismic variations which only express the lack of reproducibility of the measurement are observed. Among the many causes, the variations of the seismic signal from one survey to the next, the variability of the acquisition noises between different surveys, the imprecise position of the pickups and of the seismic sources can be mentioned.

[0012] Despite extensive reprocessing efforts to homogenize the various measurement surveys before interpretation, by means of methods described in the aforementioned publication by Ross et al. (1996), a residual non-repeatability remains, which is not insignificant. Thus, at the level of the reservoir, part of the variation of the seismic response is due to this non-reproducibility of the measurement, the other part being of course related to the physical evolutions of the reservoir as a result of the production mechanisms.

[0013] It is therefore very important, in the interpretation of the 4D measurement, to take into account this uncertainty inherent in the measurement, and not related to the reservoir variations. We therefore propose using a fuzzy discriminant analysis technique, which is the object of the aforementioned patent FR-EN-00/11,618, and applying it to the analysis of seismic events from the reservoir. The measurement uncertainties related to their imperfect reproducibility will first be evaluated using jointly seismic observations of the various surveys, made outside the zone potentially affected by the production of hydrocarbons.

SUMMARY OF THE INVENTION

[0014] The method according to the invention allows to facilitate identification of the changes, in the course of time, in the physical state of a first zone of an underground formation (a reservoir zone for example) from the changes detectable within a first time window on several seismic trace sets obtained respectively during successive seismic surveys, by taking account of the uncertainties on a certain number of descriptive seismic attributes, by reference to parts of the seismic traces of the various sets recorded in at least a second time window corresponding to at least a second zone of the underground formation (outside the reservoir) where the formation undergoes no significant physical state variation during the successive seismic surveys, wherein a discriminant analysis technique is used to classify seismic events located on the recorded traces into defined categories.

[0015] The method comprises:

[0016] forming a learning base comprising physical states that have already been recognized and classified into predetermined categories, each one being defined by attributes of known statistical characteristics,

[0017] constructing, by reference to the learning base, a classification function using a discriminant analysis technique, allowing to distribute in said categories the various seismic events to be classified from available measurements on a certain number of attributes, this function being formed by determining the probabilities of belonging of the events to the various categories by taking account of uncertainties on the attributes in form of probability intervals of variable width, and

[0018] assigning each seismic event to at least one of the predetermined categories according to the width of the probability intervals.

[0019] Said uncertainties involved in the construction of the classification function are here uncertainties expressing the lack of reproducibility of the seismic attributes from one seismic survey to the next, which are obtained by statistical analysis of the attribute variations of the seismic events of the second time window.

[0020] According to an implementation mode, the learning base is formed from seismic events measured in the vicinity of wells drilled through the formation studied, by defining therefrom learning classes corresponding to different rock natures or to different fluid contents, the various objects to be classified being associated with seismic attributes covering the formation, and for which the probability of belonging to each of the defined learning classes is evaluated in form of an interval whose boundaries depend on said seismic attributes and on the uncertainties on said attributes, these objects being assigned to at least one of the learning classes according to the relative width of the associated probability interval in relation to all of the probability intervals.

[0021] The learning base can be formed by selecting for example the seismic traces in the parts which are the most representative of the different supposed physical states of the first zone, and of their variations, obtained for example with a numerical flow and production simulation model.

[0022] The learning base can also be formed according to the modes of a multivariate probability density function calculated from all of the seismic events characterized by the selected attributes.

[0023] According to an implementation mode, the uncertainties on the seismic attributes of the first zone are estimated from the variations of the vertical mean of the attributes variations of the various seismic surveys in said second time window.

[0024] It is also possible to estimate the uncertainties on the seismic attributes in the first zone from three-dimensional stochastic simulations in order to reproduce, for the first zone, the spatial variability and statistical characteristics such as the mean and/or the variance, estimated by geostatistical analysis of the variations of the attributes in the various seismic surveys in said second time window.

[0025] According to an implementation mode, the evolution with time of the states of a system is monitored by remote sensing.

[0026] If necessary, the method can comprise preprocessing of the seismic traces so as to eliminate, on the trace parts of the successive trace sets included in the second time window, differences other than those related to said changes in the shape of said objects.

[0027] Taking account of the 4D uncertainties in the proposed interpretation process can lead to categories of the physical state of the reservoir that may be no longer recognized if the uncertainty on the measurements is too great, or to several possible categories, non-detectable as a result of the uncertainty level. The interpretation of the repeated seismic data which is thus made integrates then completely the non-reproducible aspect of this measurement type, and the random variations induced in the reservoir are no longer interpreted as physical variations of this reservoir.

BRIEF DESCRIPTION OF THE FIGURES

[0028] Other features and advantages of the method according to the invention will be clear from reading the description hereafter of a non-limitative example, with reference to the accompanying drawings wherein:

[0029]FIGS. 1a to 1 c show three seismic amplitude volumes S1 to S3 recorded at three different periods in a reservoir whose production started between acquisitions S1 and S2;

[0030]FIGS. 2a to 2 c show, for each seismic survey performed, a seismic amplitude volume located above the formation studied, corresponding to a zone where no physical change related to the production of the reservoir has occurred, and used as a basis for estimating the seismic measuring error;

[0031]FIG. 3 shows the chart of the mean horizontal variations of the seismic amplitude measurement uncertainty, corresponding to the absolute amplitude differences observed between the three data volumes of FIG. 2;

[0032]FIG. 4 shows the spatial distribution of the seismic traces selected to calibrate the imprecise classification function, coded according to their belonging class;

[0033]FIGS. 5a to 5 c show the most reliable assignments of the seismic events of the three surveys of FIG. 1, considering the uncertainties shown in FIG. 3;

[0034]FIGS. 6a to 6 c show, for these reliably assigned events, the uncertainty on the assignment probability associated with the classification;

[0035] FIGS. 7 to 9 show the spatial extension of the reliable (gray) and uncertain (black) assignments for 3 different seismic facies (facies 1: FIGS. 7a to 7 c, facies 2: FIGS. 8a to 8 c and facies 6: FIGS. 9a to 9 c), as well as the evolution of this interpretation for the three surveys.

DETAILED DESCRIPTION

[0036] The seismic events to be classified are characterised by seismic parameters or attributes. For example, these parameters can be the succession of the amplitudes along the seismic window studied. These events are simultaneously extracted from the various records obtained from the repeated seismic survey, at the level of a target zone of the subsoil, or reservoir. The class definition methodology based on the fuzzy discriminant analysis comprises four stages:

[0037] I—Analysis of the Statistical Variability of the Seismic Attributes

[0038] The first stage consists in analysing the statistical variability of the seismic attributes between the recording operations of the 4D seismic survey, due to the imperfect reproducibility of the measurement in the course of time. We therefore consider on each seismic survey a volume of data taken in the time window for which prior reprocessing of the data has been carried out, but far enough from the reservoir zone so that the variations observed cannot be attributed to the physical evolutions of the reservoir, related to the production mechanisms. The different seismic data volumes are characterised by a set of parameters or attributes that will be used for classifying the seismic events. The variations, from one survey to the next, of the attributes considered on the events associated with all of these data are then studied. A parameter is defined thereafter, which represents the variability as a function of time, and therefore of production, of the seismic attributes considered. This parameter may be, for example, the absolute or relative difference of seismic attributes between the various seismic surveys, for each time sample and each seismic event that constitute the measurement uncertainty analysis volume. The statistical characteristics and the spatial variability of this parameter representing the uncertainty are then described in order to be able to extrapolate it between the uncertainty evaluation zone and the reservoir zone. For exampe, in case of a vertical stationarity of the uncertainty, one may assume that the seismic measuring error in the reservoir is also vertically stationary, and that the vertical mean calculated on the uncertainty evaluation zone is representative of the vertical mean of this error in the reservoir.

[0039] II—Definition of the Learning Classes

[0040] The second stage is a stage of definition of the learning classes to be used in the discriminant analysis. This stage is carried out by indicating, among all the surveys, the seismic events supposed to be representative of the various classes of the reservoir studied.

[0041] Various methods can be used therefore. Two of them are mentioned hereafter by way of example.

[0042] A first possibility consists in extracting the seismic events recorded in the vicinity of wells, whose geologic interpretation (and the evolution in the course of time of this interpretation) has been carried out. This methodology will then allow to classify all of the seismic traces according to geologic variations observed in the wells. A second possibility could consist in using the assumed physical state variations of the reservoir, obtained for example with a flow and production simulation model, and in extracting the seismic learning events from the zones that are the most representative of these assumed states. Another possibility consists in carrying out a non-supervised classification of the seismic events recorded during the various surveys at different times, and in using the classes thus formed as learning classes in the fuzzy discriminant analysis.

[0043] III—Calibration of a Fuzzy Classification Function

[0044] The third stage consists in calibrating by fuzzy discriminant analysis a fuzzy classification function from the seismic events selected in the previous stage so as to represent the object classes considered, and the uncertainties related thereto, estimated in stage 1. The method used, described in the aforementioned patent application FR-EN-00/11,618, and reminded hereafter for the sake of clarity, is therefore applied to the uncertain data.

[0045] In its broad lines, this method comprises constructing a function allowing to classify objects into predetermined categories. This function is elaborated from the statistical characteristics of the attributes describing already recognized objects (i.e. for which the belonging category is known), which form a learning set. The classification function thus constructed is based on the calculation of the probabilities of belonging of the objetcs to the various classes, considering the measurements of p attributes available on these objects. One may for example decide to assign the object to the category for which its belonging probability is the most probable. The uncertainties on the attributes are taken into account in form of an interval of possible variation of the measurement of each attribute on a certain object. The aim is to propagate these possible intervals, for the values of the various measurements, in the classification probabilities calculation. For each category considered, we thus obtain a probability interval instead of a probability of classifying the object into this category. The object can then be assigned from the moment that one of these probability intervals exceeds the others. Analysis of these probability intervals also allows to evaluate the quality of prediction of the learning categories, as well as the degree of separation of these categories provided by the attributes, while integrating the uncertainty on the experimental measurements of these attributes.

[0046] This fuzzy discriminant analysis method is based on a particular application of the concept of interval arithmetic, reminded hereafter to facilitate comprehension of the description.

[0047] We therefore use the technique referred to as interval arithmetic, described in the aforementioned publication by Moore R. E., 1969, which allows to extend the usual mathematical operators to calculations on intervals. The aim is to provide a guaranteed frame for the results considering the input intervals. Thus, rules (1) hereafter define the extension of the arithmetic operations to two intervals x=[x⁻; x⁺] and y=[y⁻; y⁺]. $\begin{matrix} \left\{ \begin{matrix} {{x + y} = \left\lbrack {{x^{-} + y^{-}};{x^{+} + y^{+}}} \right\rbrack} \\ {{x - y} = \left\lbrack {{x^{-} - y^{+}};{x^{+} - y^{-}}} \right\rbrack} \\ \left. {{x \cdot y} = \left\lbrack {{{Min}\left\{ {{x^{-} \cdot y^{-}};{x^{-} \cdot y^{+}};{x^{+} \cdot y^{-}};{x^{+} \cdot y^{+}}} \right\}};{{Max}\left\{ {{x^{-} \cdot y^{-}};{x^{-} \cdot y^{+}};{x^{+} \cdot y^{-}};{x^{+} \cdot y^{+}}} \right\}}} \right.} \right\rbrack \\ {\frac{1}{x} = \left\lbrack {\frac{1}{x^{+}};\frac{1}{x^{-}}} \right\rbrack} \\ {\frac{x}{y} = {x \cdot \frac{1}{y}}} \end{matrix} \right. & (1) \end{matrix}$

[0048] For any function f, Equation (2) defines its extension to the intervals, referred to as inclusion function ƒ_([ ]).

ƒ_([ ]() x){y=f(x)|xεx}  (2)

[0049] Obtaining these inclusion functions generally poses no problem. Among these, the natural inclusion function is constructed using only calculation rules (1) and some additional definitions concerning the elementary functions. For example, Equation (3) defines the extension of the exponential:

exp_([ ]() x)=└exp(x ⁻);exp(x ⁺)┘  (3)

[0050] However, the natural inclusion functions are generally not optimum insofar as inclusion (2) is not an equality. The objective of interval arithmetic is then to generate an inclusion function whose boundaries are as limited as possible.

[0051] To complete these basic definitions, we define an extension of the comparison operators to the intervals [equation (4)].

x>y

⇄x ⁻ >y ⁺  (4)

[0052] It can be noted that the latter definition allows to compare disjointed intervals. Intervals that overlap one another are referred to as indissociable.

[0053] The concepts of interval arithmetic are applied in the fuzzy discriminant analysis method to frame probabilistic objects. Now, probability intervals cannot strictly verify axioms referred to as Kolmogorov axioms which define a probability and are published in the aforementioned document by Kolmogorov (1950). It is therefore necessary to generalize the probability theory to the intervals. This generalization is described by Walley in the aforementioned publication under the name of imprecise probability theory. The two principal axioms to be verified by an imprecise probability p_([ ]) are reminded hereafter.

[0054] p_([ ]) is a positive defined measurement, i.e., for any event A

0≦p⁻ _([ ](A)≦p) ⁺ _([ ])(A)≦1  (5)

[0055] p_([ ]) verifies a coherence axiom, i.e., for any set of independent events A_(i), there is a function p defined on this set of events, which verifies the Kohnogorov axioms, and such that, for all the A_(i),

p⁻ _([ ])(A_(i))≦(A_(i))≦p⁺ _([ ])(A_(i))  (6)

[0056] The object recognition method described hereafter is similar, in its broad lines, to a discriminant analysis algorithm

[0057] In the description below, one of the N predetermined categories is denoted by C_(i). The jth learning interval of class C_(i), consisting of a vector of p measurement intervals, is denoted by x_(ij)=(x_(ij) ⁽¹⁾; . . . ; x_(ij) ^((k)); x_(ij) ^((p))). The current interval of the attribute space is denoted by x=(x⁽¹⁾; . . . ; x^((k)), . . . ; x^((p))). Finally, x^(c) denotes the centre of any interval x.

[0058] The stages of the object recognition algorithm are:

[0059] III-1 Calculation of the Conditional Probability Densities p_([ ])(X/C_(i))

[0060] The probability density can be estimated using either a non-parametric method or a parametric method. In the first case, the advantage of the method is that it allows better identification of the structure of each learning class C_(i). However, its use requires a sufficient size for learning class C_(i) so as to allow reliable identification of this structure. In the opposite case, it is necessary to make an assumption on the structure of learning class C_(i). This amounts to supposing that this class follows a predetermined distribution law, a Gaussian law for example [Equation (7)]: $\begin{matrix} {{p\left( {x/C_{i}} \right)} = {\frac{1}{\left( {2\pi} \right)^{P/2}{\sum }^{1/2}}^{{- \frac{1}{2}}{({x - \mu})}^{t}{\sum^{- 1}{({x - \mu})}}}}} & (7) \end{matrix}$

[0061] where μ represents the centre of inertia of learning class C_(i) and Σ its variance-covariance matrix characteristic of its dispersion.

[0062] We successively describe the extensions of the non-parametric method for estimating the probability density, then of the Gaussian parametric method [Equation (7)].

[0063] III-1a Non-Parametric Method

[0064] In the non-parametric method, we estimate the conditional density, for example by means of the kernel method. The aim is to apply to the intervals the formula for calculating the conditional probability density function by means of Epanechnikov's kernel method described in the aforementioned reference: $\begin{matrix} {{\left. {{{p_{\lbrack\quad\rbrack}\left( x \right.}}C_{i}} \right) = {\frac{1}{n_{j}h^{p}}{\sum\limits_{j = 1}^{n_{1}}{K_{\lbrack\quad\rbrack}\left( \frac{x - x_{ij}}{h} \right)}}}},} & (8) \end{matrix}$

[0065] where h represents the height of the passband of the kernel, n_(i) the size of learning class C_(i). The kernel is written as follows: $\begin{matrix} {{K_{\lbrack\quad\rbrack}\left( \frac{x - x_{ij}}{h} \right)} = \left\{ \begin{matrix} {{\frac{1}{2N_{p}}\left( {p + 2} \right)\left( {1 - \frac{\sum\limits_{k = 1}^{p}\left( {x^{(k)} - x_{ij}^{(k)}} \right)^{2}}{h^{2}}} \right)\quad {if}\quad {{x - x_{ij}}}} < h} \\ {0\quad {otherwise}} \end{matrix} \right.} & (9) \end{matrix}$

[0066] Each quadratic term of the sum is independent of the others. We give here the expression of the lower and upper boundaries of these terms Q=(x^((k))−x^((k))ij)/h². $\begin{matrix} \left\{ \begin{matrix} {Q^{-} = \left\{ \begin{matrix} {{\frac{\left( {x^{c{(k)}} - x_{ij}^{{(k)} -}} \right)^{2}}{h^{2}}\quad {if}\quad x_{ij}^{{(k)}c}} \leq x^{{(k)}c} \leq {x_{ij}^{{(k)} -} - \left( {x^{{(k)}c} - x^{{(k)} -}} \right) + h}} \\ {{{\frac{\left( {x^{c{(k)}} - x_{ij}^{{(k)} +}} \right)^{2}}{h^{2}}\quad {if}\quad x_{ij}^{{(k)} +}} - \left( {x^{{(k)} +} - x^{{(k)}c}} \right) - h} \leq x^{{(k)}c} \leq x_{ij}^{{(k)}c}} \\ {1\quad {otherwise}} \end{matrix} \right.} \\ {Q^{+} = \left\{ \begin{matrix} {{{\frac{\left( {x_{ij}^{{(k)} -} - x^{{(k)} -}} \right)^{2}}{h^{2}}\quad {if}\quad x_{ij}^{{(k)} -}} - \left( {x^{{(k)}c} - x^{{(k)} -}} \right) - h} \leq x^{{(k)}c} \leq {x_{ij}^{{(k)} -} - \left( {x^{{(k)}c} - x^{{(k)} -}} \right)}} \\ {{{0\quad {si}\quad x_{ij}^{{(k)} -}} - \left( {x^{{(k)}c} - x^{{(k)} -}} \right)} \leq x^{{(k)}c} \leq {x_{ij}^{{(k)} +} + \left( {x^{{(k)} +} - x^{{(k)}c}} \right)}} \\ {{{\frac{\left( {x^{{(k)} +} - x_{ij}^{{(k)} +}} \right)^{2}}{h^{2}}\quad {if}\quad x_{ij}^{{(k)} +}} + \left( {x^{{(k)} +} - x^{{(k)}c}} \right)} \leq x^{{(k)}c} \leq {x_{ij}^{{(k)} +} - \left( {x^{{(k)} +} - x^{{(k)}c}} \right) + h}} \\ {1\quad {otherwise}} \end{matrix} \right.} \end{matrix} \right. & (10) \end{matrix}$

[0067] An equivalent calculation would be carried out if another non-parametric estimator of the conditional density were used, such as the estimator of the k closest neighbours.

[0068] III-1b Parametric Method

[0069] Equation (7) can theoretically be extended by means of calculation rules (1), but their direct use leads to overestimate the variation intervals of the probability densities of Equation (11). $\begin{matrix} {{p_{\lbrack\quad\rbrack}\left( {x_{\lbrack\quad\rbrack}/C_{i}} \right)} = {\frac{1}{\left( {2\pi} \right)^{P/2}{\sum_{\lbrack\quad\rbrack}}^{1/2}}^{{- \frac{1}{2}}{({x_{\lbrack\quad\rbrack} - \mu_{\lbrack\quad\rbrack}})}^{t}{\sum_{\lbrack\quad\rbrack}^{- 1}{({x_{\lbrack\quad\rbrack} - \mu_{\lbrack\quad\rbrack}})}}}}} & (11) \end{matrix}$

[0070] The calculation algorithm proposed here allows to improve the frame that could be obtained by applying calculation rules (1). The various stages thereof are:

[0071] Calculation of the Variation Intervals of Parameters μ and Σ of Gaussian Law (11)

[0072] This calculation consists in finding the minimum and the maximum for each term of matrix Σ when points x_(ij) of learning class C_(i) vary within their possible values interval x_(ij). It is carried out using an optimization method under constraint such as the projected gradient method. $\begin{matrix} \left\{ {{\begin{matrix} {\overset{-}{\sum\limits_{kl}}{= {\min\limits_{x_{ij} \in \quad {x_{{ij},}{\forall j}}}\left\{ {\sum\limits_{j}{\left( {x_{ij}^{(k)} - \mu_{i}^{(k)}} \right)\left( {x_{ij}^{(I)} - \mu_{i}^{(I)}} \right)}} \right\}}}} \\ {\overset{+}{\sum\limits_{kl}}{= {\max\limits_{x_{ij} \in \quad {x_{{ij},}{\forall j}}}\left\{ {\sum\limits_{j}{\left( {x_{ij}^{(k)} - \mu_{i}^{(k)}} \right)\left( {x_{ij}^{(I)} - \mu_{i}^{(I)}} \right)}} \right\}}}} \end{matrix}{\forall\left( {k,1} \right)}} = {1\quad \ldots \quad p}} \right. & (12) \end{matrix}$

[0073] Diagonalization of the Interval Matrix Σ_([ ])

[0074] This stage consists in framing interval matrix Σ_([ ]) by a matrix Σ*_([ ]) that is similar thereto, but diagonal. In other words, Σ*_([ ]) must meet equation (13):

Σ*_([ ]) ⊃R^(t) _(θ)Σ_([ ])R_(θ)  (13)

[0075] where R₇₄ is a rotation matrix.

[0076] We first modify matrix Σ_([ ]) by trying to convert it into a matrix Σ′_([ ])=R^(t) _(θ)Σ_([ ])R_(θ whose out-of-diagonal terms vary within intervals that are as small as possible. We therefore use Jacobi's interval method. We then replace the out-of-diagonal intervals of Σ′) _([ ]) by 0. This operation necessarily leads to an increase in the size of the variation intervals of the diagonal terms of Σ′_([ ]).

[0077] In short, after this second stage, we have found a frame for matrix Σ_([ ]) in the form of a matrix Σ*_([ ]), and we have therefore overcome interval repetition problems. However, direct use of matrix Σ*_([ ]) in Equation (11) still leads to overestimate the variation intervals of the conditional probability densities.

[0078] Optimization of the Variation Intervals of the Conditional Probability Densities

[0079] In order to better frame the variation interval of the conditional probability densities, we evenly divide the variation domain of μ into n_(s) subdomains μ_(k[ ]). In each subdomain μ_([ ]) thus formed, we apply interval arithmetic rules (1). This operation provides an interval function p_(k[ ]) (x/C_(i)). After forming the n_(s) interval functions, we calculate function p_([ ]) (x/C_(i)) which is the union of all the interval functions p_(k[ ])(x_([ ])/C_(i)) previously formed: $\begin{matrix} {{p_{\lbrack\quad\rbrack}\left( {x/C_{i}} \right)} = {\bigcup\limits_{k}{p_{k{\lbrack\quad\rbrack}}\left( {x/C_{i}} \right)}}} & (14) \end{matrix}$

[0080] Function p_([ ])(x_([ ])/C_(i)) thus calculated is a guaranteed frame for the variation intervals of the conditional probability density, but whose boundaries are better than if no subdomains had been formed.

[0081] After calculating the variation intervals of the conditional probability densities, we calculate the a posteriori probabilities p_([ ])(C_(i)/x).

[0082] III-2 Calculation of the a Posteriori Probabilities p_([ ])(C_(i)/x)

[0083] In this stage, we apply to the intervals the Bayes rule well-known in statistics: $\begin{matrix} {\left. {{p_{{\lbrack\quad\rbrack}\quad}\left( C_{i} \right.}x} \right) = \frac{\left. {{{p\left( x \right.}}C_{i}} \right) \cdot {p\left( C_{i} \right)}}{\left. {\sum\limits_{i = 1}^{p}{{{p\left( x \right.}}C_{i}}} \right) \cdot {p\left( C_{i} \right)}_{\lbrack\quad\rbrack}}} & (15) \end{matrix}$

[0084] By converting the previous equation and by applying rules (1), we obtain the optimum expression hereafter for the a posteriori probabilities: $\begin{matrix} \left. {{\left. {{{p_{\lbrack\quad\rbrack}\left( C_{i} \right.}}x} \right) = \left\lbrack \left( {1 + {\sum\limits_{l \neq i}\frac{\left. {{{p^{+}\left( x \right.}}C_{i}} \right) \cdot {p^{+}\left( C_{i} \right)}}{\left. {{{p^{-}\left( x \right.}}C_{i}} \right) \cdot {p^{-}\left( C_{i} \right)}}}} \right)^{- 1} \right.};\left( {1 + {\sum\limits_{l \neq i}\frac{\left. {{{p^{-}\left( x \right.}}C_{i}} \right) \cdot {p^{-}\left( C_{i} \right)}}{\left. {{{p^{+}\left( x \right.}}C_{i}} \right) \cdot {p^{+}\left( C_{i} \right)}}}} \right)^{- 1}} \right\rbrack & (16) \end{matrix}$

[0085] These interval probabilities verify the imprecise probability axioms.

[0086] III-3 Classification of Interval x in the Likeliest Class or Classes

[0087] The classification mode used here is an extension of the maximum likehood rule. It consists in comparing the various a posteriori probability intervals p_([ ])(C_(i)/x). The various intervals are thus first arranged in descending order of p⁺(C_(i)/x) or, which is equivalent, in descending order of quantities p⁺(x/C_(i))p⁺(C_(i)):

p⁺(x|C_(i1))p⁺(C_(i1))≧p⁺(x|C_(i2))p⁺(C_(i2))≧ . . . ≧p⁺(x|C_(iN))p⁺(C_(iN))  (17)

[0088] Then, by applying the rule of comparison on the intervals, it follows that, if intervals p_([ ])(C_(i1)/x) and p_([ ])(C_(i2)/x) are disjointed (p⁻(C_(i1)/x)≧p⁺(C_([ ])/x)), interval x is assigned to class C_(i1). In the opposite case, the algorithm cannot distinguish classes C_(i1) and C_(i2) at x. The previous comparison test is then repeated between classes C_(i1) and C_(i3), . . . , C_(i1) until intervals p_([ ])(x/C_(i1)).p_([ ])(C_(i1)) and p_([ ])(x/C_(i1)).p_([ ])(C_(i1)) are disjointed.

[0089] This stage has allowed to calibrate on the learning classes a classification function integrating the measurement uncertainties.

[0090] IV—Classification of the Seismic Events

[0091] Once this imprecise classification function calibrated, it is used to classify the (also imprecise) seismic events of all the surveys. For each imprecise seismic event, the interval of possible variations of the probability of assignment to each class is calculated. According to the relative size of these various intervals, notably according to their possible overlapping, the seismic event is assigned to a class set compatible with the measured attributes and the uncertainties related thereto.

[0092] V—Applications

[0093] In order to monitor the physical changes in the reservoir related to the production mechanisms, three seismic surveys have been recorded: the first one before producing a reservoir, and the other two several months after production start. These changes are analysed within a constant time window shown in FIGS. 1a to 1 c. The seismic events analysed are the seismic trace portions that can be extracted from each of these data volumes, and the attributes used to represent them are the amplitudes sampled over the 11 successive time intervals included in the reservoir window.

[0094] In order to estimate the uncertainties related to the lack of reproducibility of the data, we have also extracted from the three seismic records a time window located way above the reservoir, but in the zone that has been subjected to reprocessing before interpretation (FIGS. 2a to 2 c). The amplitude measurement variations observed in this zone are not linked with the production mechanisms.

[0095] The variations between FIGS. 2a to 2 c are synthesized in FIG. 3, which represents the vertical mean of the maximum variation of the seismic amplitudes between the various surveys, and for each seismic event. In the illustrative example given, this FIG. 3 is considered to represent the vertically stationary measurement uncertainty on each seismic amplitude for the whole data block (therefore including the reservoir window). This uncertainty varies laterally as shown in the chart of FIG. 3 (horizontal non-stationarity).

[0096] Once the measurement uncertainties evaluated, the classes are defined from the analysis of the peaks of the multivariate probability density function calculated on all of the seismic traces of the three surveys. The seismic traces of higher probability density forming these peaks are then selected for learning of the classification function. FIG. 4 shows the spatial distribution of these learning seismic traces for the three surveys.

[0097] An imprecise classification function is then calibrated from the learning seismic traces of FIG. 4 bearing a measurement uncertainty shown in FIG. 3.

[0098] This function is applied in a last stage to interpret the change in the spatial distribution of the object classes previously defined in the reservoir. These changes will then be interpreted as physical changes related to the production mechanisms. FIGS. 5a to 5 c show, for each survey, the assignments referred to as stable, i.e. the seismic events for which the measurement uncertainty has no effect on the classification result. These are all the points that are not coded in white. Globally, the majority of the seismic events are classified in a stable way. For the latter, the uncertainty on the corresponding classification probability is generally very low, as shown in FIGS. 6a to 6 c, which reinforces the feeling of reliability of the classification. However, it can be seen in FIGS. 5b and 5 c that the classification is uncertain (or non-single) in the southern part of the reservoir. If we observe more precisely the possible assignment charts for the various facies, we see that these points are classified either in class 1 or in class 2, as shown in FIGS. 7b and 7 c, and in FIGS. 8b and 8 c. On the other hand, it can simultaneously be seen that facies 6 (FIGS. 9b and 9 c), which had been reliably located before production start (FIG. 9a), has disappeared. An interpretation of this result is that facies 6 disappears as a result of the production of the reservoir since the seismic characteristics of the zones where it was represented evolve. 

1) A method for facilitating identification of changes occurring, in the course of time, in the physical state of a first zone of an underground formation, from changes detectable within a first time window on several sets of seismic traces respectively obtained during successive seismic surveys, by taking account of uncertainties on a certain number of descriptive seismic attributes, by reference to parts of said traces of the various sets recorded in at least a second time window corresponding to at least a second zone of the underground formation where said formation undergoes no significant physical state variation during the successive seismic surveys, wherein a discriminant analysis technique is used to classify seismic events located on the record traces into defined categories, the method comprising: forming a learning base comprising physical states that have already been recognized and classified into predetermined categories, each one being defined by attributes of known statistical characteristics, constructing, by reference to the learning base, a classification function using a discriminant analysis technique, allowing to distribute in said categories the various seismic events to be classified from available measurements on a certain number of attributes, this function being formed by determining the probabilities of belonging of the events to the various categories by taking account of uncertainties on the attributes in form of probability intervals of variable width, and assigning each seismic event to at least one of the predetermined categories according to the width of the probability intervals, the method being characterised in that said uncertainties involved in the construction of the classification function are uncertainties expressing the lack of reproducibility of the seismic attributes from one seismic survey to the next, which are obtained by statistical analysis of the attribute variations of the seismic events of the second time window. 2) A method as claimed in claim 1, wherein the learning base is formed from seismic events measured in the vicinity of wells drilled through the formation studied, by defining therefrom learning classes corresponding to different rock natures or different fluid contents, the various seismic events to be classified being associated with seismic attributes covering the formation and for which the probability of belonging to each of the learning classes defined is evaluated in form of an interval whose boundaries depend on said seismic attributes and on the uncertainties on these attributes, and these seismic events are assigned to at least one of the learning classes according to the relative width of the associated probability interval in relation to all of the probability intervals. 3) A method as claimed in claim 1, characterised in that the learning base is formed by selecting the seismic traces in the parts which are the most representative of the supposed different physical states of the first zone and of their variations, obtained for example with a numerical flow and production simulation model. 4) A method as claimed in claim 1, characterised in that the learning base is formed according to the modes of a multivariate probability density function calculated from all the seismic events characterised by the attributes selected. 5) A method as claimed in any one of claims 1 to 4, characterised in that the uncertainties on the seismic attributes of the first zone are estimated from the variations of the vertical mean of the attribute variations of the various seismic surveys, in said second time window. 6) A method as claimed in any one of claims 1 to 4, characterised in that the uncertainties on the seismic attributes of the first zone are estimated from three-dimensional stochastic simulations in order to reproduce, for the first zone, the spatial variability and statistical characteristics, such as the mean and/or the variance, estimated by geostatistical analysis of the attribute variations in the various seismic surveys, in said second time window. 7) A method as claimed in any one of claims 1 to 6, wherein the evolution in the course of time of the states of a system is monitored by remote sensing. 8) A method as claimed in any one of the previous claims, characterised in that it comprises preprocessing the seismic traces so as to eliminate, on the trace portions of the successive trace sets included in the second time window, the differences other than those related to said changes in the shape of said seismic events. 